Speed is a measure of distance in a certain time, or distance divided by time. With this, you can get an equation relating the three, like speed = distance/time. If you rearrange this equation, it can become distance = speed*time, or time = distance/speed.

(a) calculate the distance travelled by bernard in this time
You know speed and time, so you can plug them into the first equation to find distance, so:
distance = 200*1.5 = 300km

(b) how long does it take keith to finish the race?
You now know distance and speed, so you can find time with the second equation:
time = 300/160 = 1.875 hours

You have to be careful when you use these equations, though, and make sure that the units of time and speed match up. Since the speed was km per hour, and our time was hours, we were ok.